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Discrete Subsets of Proximity Spaces

Published online by Cambridge University Press:  20 November 2018

Don A. Mattson
Don A. Mattson*
Affiliation:
Moorhead State University, Moorhead, Minnesota
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The distinct Hausdorff compactifications δX of a completely regular (Hausdorff) space X are in one-one correspondence with the admissible proximity relations δ on X, or alternatively, with the admissible totally bounded uniform structures for X. (See [1], [2].) Thus, δX is the Smirnov compactification of (X, δ). Generalized uniform structures for X will be described by means of pseudometrics on X (cf. [5], [7], [13]). Let where is in the proximity class π(δ) associated with (X, δ). Then a subset S of X is σ-discrete of gauge if , for all x, y ∈ S where x ≠ y.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 225 - 230
Copyright
Copyright © Canadian Mathematical Society 1979

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